In space with respect to an orthonormal coordinate system $( \mathrm { O } , \vec { \imath } , \vec { \jmath } , \vec { k } )$, we consider the points
$$\mathrm { A } ( - 1 ; - 1 ; 3 ) , \quad \mathrm { B } ( 1 ; 1 ; 2 ) , \quad \mathrm { C } ( 1 ; - 1 ; 7 )$$
We also consider the line $\Delta$ passing through the points $\mathrm { D } ( - 1 ; 6 ; 8 )$ and $\mathrm { E } ( 11 ; - 9 ; 2 )$.

\begin{enumerate}
  \item a. Verify that the line $\Delta$ admits the following parametric representation:
$$\left\{ \begin{aligned}
x & = - 1 + 4 t \\
y & = 6 - 5 t \quad \text { with } t \in \mathbb { R } \\
z & = 8 - 2 t
\end{aligned} \right.$$
b. Specify a parametric representation of the line $\Delta ^ { \prime }$ parallel to $\Delta$ and passing through the origin O of the coordinate system.\\
c. Does the point $\mathrm { F } ( 1.36 ; - 1.7 ; - 0.7 )$ belong to the line $\Delta ^ { \prime }$?
  \item a. Show that the points $\mathrm { A }$, $\mathrm { B }$ and $\mathrm { C }$ define a plane.\\
b. Show that the line $\Delta$ is perpendicular to the plane (ABC).\\
c. Show that a Cartesian equation of the plane (ABC) is: $4 x - 5 y - 2 z + 5 = 0$.
  \item a. Show that the point $\mathrm { G } ( 7 ; - 4 ; 4 )$ belongs to the line $\Delta$.\\
b. Determine the coordinates of the point H, the orthogonal projection of point G onto the plane (ABC).\\
c. Deduce that the distance from point G to the plane (ABC) is equal to $3 \sqrt { 5 }$.
  \item a. Show that the triangle ABC is right-angled at A.\\
b. Calculate the volume $V$ of the tetrahedron ABCG.\\
We recall that the volume $V$ of a tetrahedron is given by the formula $V = \frac { 1 } { 3 } \times B \times h$ where B is the area of a base and h the height corresponding to this base.
\end{enumerate}